Firn Densification

Monday: Introduction to basic concepts, Herron and Langway model

(Everything seen through the scope of the relevance for gas and water isotope studies. So we are going to jump a little bit back and for...if you get lost stop me and ask..!)

Tuesday: Coding, coding, coding and exercise

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Image("../figs/bubbles_ice_core.png", width = 650)
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Why is it important to describe densification?

Past Atmosphere reconstructions - $\Delta_{age}$

Gas age distributions

Gas thermometry

Water isotope diffusion - Layer counting

Water isotope diffusion as a thermometer

Elevation changes - Mass balance estimates

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Image("../figs/2580_02.png", width = 100)
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Image("../figs/2580_02det.png", width = 850)
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How do bubbles look before they become bubbles? (video)

Interconnected air channels allowing diffusive transport

Gradual transformation from snow to ice

Amount of channels decreases with depth until a point where bubbles are occluded

Stages of densification

  • Initial snow to critical density $\left( \rho<550 {\text{ kgm}}^{-3} \right)$ There is rapid densification at this stage via grain settling and packing. Particles rapidly reduce the total surface area of the crystals
  • Critical Density to approximately close-off depth Densification rates are slower at this stage with density slowly reaching close-off values $\left(550<\rho<820 \text{ kgm}^{-3} \right)$. At close off the air passages become clossed-off and form air bubbles.
  • Close-off density to ice $\left(820<\rho<920 \text{ kgm}^{-3} \right)$ At this stage about 90% of the air has ascaped to the surface and the individual bubbles compress further for the ice to reach $920 \text{ kgm}^{-3}$

The process of densification is temperature and accumulation dependant.

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Image("../figs/dome_c_density_herron.png", width = 700)
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Some values to remember:

Description of medium Density Range $\left[ \text{kgm}^{-3} \right]$
New snow 50-70
Settled snow 200-300
Wind packed snow 350-400
Firn 400-830
Glacial Ice 830-920

Some definitions

Porosity

$s = 1-\frac{\rho}{\rho_{\text{ice}}}$

Porosity is the interstitial space between ice crystals. It can be referred to as open porosity if pores are still connected with the atmosphere, or closed if pores have been closed-off.

Mean close-off density

${\overline{\rho}}_{co} = {\left( \frac{1}{\rho_{ice}} + 6.95 \times 10^{-7}T-4.3 \times 10^{-5} \right)}^{-1} $

The average density at which the bubbles are close-off. Mostly dependant on temperature (and wind).

Full close-off depth

The depth at which all air has been occluded into bubbles.

Lock-in depth

The depth at which vertical gas transport has ceased. Likely due to high-density layers above the close-off depth. There is still open porosity below this depth and often it can be pumped

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Image("../figs/buizert2013_densif_stages.png", width = 700)
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Image("../figs/herron_title.png", width = 800)
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Basic assumption for the H-L model is that the proportional changes of density are linearly related to the overburden load, thus:

$\frac{d\rho}{\rho_i - \rho} = \Gamma\rho dh$

Exercise:

Prove the linear relationship between $\ln \left[\frac{\rho}{\rho_i - \rho} \right]$ and depth.

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rho_ice = 917.0
data_ngrip = np.genfromtxt("./NGRIP_density.txt", delimiter=",", skip_header=2)
z_ngrip = data_ngrip[:,0]
rho_ngrip = data_ngrip[:,1]
logrho_ngrip = np.log(rho_ngrip/(rho_ice - rho_ngrip))
plt.figure(2)
plt.plot(rho_ngrip, z_ngrip, "r.")
plt.twiny()
plt.plot(logrho_ngrip, z_ngrip, ".")
plt.ylim([120,0])
plt.ylabel("Depth [m]")
plt.xlabel("Density [kgm-3]")
plt.title("NGRIP")
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<matplotlib.text.Text at 0x10e96f990>

Lets calculate:

Critical Depth

Age at critical depth

$\rho(z) \text{ and } \rho(t)$ for upper and lower stage

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Image("../figs/grip_herron.png", width = 800)
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An overview of ice core sites characteristics is given here from Buizert et al 2013

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Image("../figs/buizer2013_sites.png", width = 900)
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Exercise

In Absalon you will find the following data sets:

  • Dome Concordia, East Antarctica (DC-dens-JP.csv)
  • GRIP, Central Greenland (grip_density.txt)
  • NEEM2007, NE Greenland (NEEM07density.txt)
  • NorthGRIP, Central Greenland (NGRIP_density.txt)
  • WAIS-D, West Antarctica (wais_density_Breton.txt)

Use the boundary conditions given in the table of Buizert et al to run the H-L model for every site in the list. Calculate the close-off depth and the age of the bubbles at this depth.

Your results are unlikely to fit every core site very well. Implement fudge parameters f0, f1 in your code in front of the activation energies and tune them so you fit the data sets in a least squares sense.

Take home exercise

In the figure below you are given a plot of measured density versus depth for a shallow core from North Greenland. Use the Herron and Langway model in order to model the density by optimizing the fudge factors in the fron of the activation energy coeeficients $K_o, K_1$

When you have a pair of fudge coefficients $f_0, f_1$ calculate the following:

  • The close-off depth $\left( \rho_{co} = 820 \text{ kgm}^{-3} \right)$.
  • The $\Delta_{\text{age}}$ in years for present conditions.
  • Calculate close-off depth and $\Delta_{\text{age}}$ for glacial conditions assuming $T=-50 {}^{\circ}$ C and $A = 5\text{ cmyr}^{-1}$
  • Create a diagram where x is density, y is depth and plot the given data, the H-L model for present conditions fitting the data and the H-L model for glacial conditions. Remember, axes labels, legents, titles etc. Make it look publication quality.

In the second figure you can see the isotopic profile for the NEEM ice core. The $\delta^{18}\text{O}$ are found in the file neem_megafile.txt uploaded in absalon. In the file you can also find information on the age, annual layer thickness and accumulation at every depth with a resolution of 55 cm.

  • Using th isotopic profile and assuming an isotope - temperature sensitivity $\frac{\text{d}\delta^{18}\text{O}}{\text{d}T} = 0.67 \text{ permile C}^{-1}$, calculate the past temperature for all the core. Present your results versus depth as well as verus age.

  • Use the accumulation column in the neemmegafile and combined with the temperatures you calculated in the previous step calculate the close-off depth as well as $\Delta{\text{age}}$ for every point of the core. Present your results versus depth and age.

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Image("../figs/neemS2007_density.png", width = 900)
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Image("../figs/neem_bags_d18.png", width = 950)
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